Step | Hyp | Ref
| Expression |
1 | | vex 3412 |
. . . 4
⊢ 𝑤 ∈ V |
2 | | oveq1 7220 |
. . . . . . . 8
⊢ (𝑣 = 𝑎 → (𝑣 · 𝑏) = (𝑎 · 𝑏)) |
3 | 2 | eqeq2d 2748 |
. . . . . . 7
⊢ (𝑣 = 𝑎 → (𝑧 = (𝑣 · 𝑏) ↔ 𝑧 = (𝑎 · 𝑏))) |
4 | 3 | rexbidv 3216 |
. . . . . 6
⊢ (𝑣 = 𝑎 → (∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏) ↔ ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 · 𝑏))) |
5 | 4 | cbvrexvw 3359 |
. . . . 5
⊢
(∃𝑣 ∈
𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 · 𝑏)) |
6 | | eqeq1 2741 |
. . . . . 6
⊢ (𝑧 = 𝑤 → (𝑧 = (𝑎 · 𝑏) ↔ 𝑤 = (𝑎 · 𝑏))) |
7 | 6 | 2rexbidv 3219 |
. . . . 5
⊢ (𝑧 = 𝑤 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 · 𝑏) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑤 = (𝑎 · 𝑏))) |
8 | 5, 7 | syl5bb 286 |
. . . 4
⊢ (𝑧 = 𝑤 → (∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑤 = (𝑎 · 𝑏))) |
9 | | supmul.1 |
. . . 4
⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑣 · 𝑏)} |
10 | 1, 8, 9 | elab2 3591 |
. . 3
⊢ (𝑤 ∈ 𝐶 ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑤 = (𝑎 · 𝑏)) |
11 | | supmul.2 |
. . . . . . . . . . 11
⊢ (𝜑 ↔ ((∀𝑥 ∈ 𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥))) |
12 | 11 | simp2bi 1148 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
13 | 12 | simp1d 1144 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
14 | 13 | sselda 3901 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ) |
15 | 14 | adantrr 717 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ ℝ) |
16 | | suprcl 11792 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈
ℝ) |
17 | 12, 16 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈
ℝ) |
18 | 17 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → sup(𝐴, ℝ, < ) ∈
ℝ) |
19 | 11 | simp3bi 1149 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)) |
20 | 19 | simp1d 1144 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ⊆ ℝ) |
21 | 20 | sselda 3901 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ ℝ) |
22 | 21 | adantrl 716 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ ℝ) |
23 | | suprcl 11792 |
. . . . . . . . 9
⊢ ((𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥) → sup(𝐵, ℝ, < ) ∈
ℝ) |
24 | 19, 23 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → sup(𝐵, ℝ, < ) ∈
ℝ) |
25 | 24 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → sup(𝐵, ℝ, < ) ∈
ℝ) |
26 | | simp1l 1199 |
. . . . . . . . . . 11
⊢
(((∀𝑥 ∈
𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)) → ∀𝑥 ∈ 𝐴 0 ≤ 𝑥) |
27 | 11, 26 | sylbi 220 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 0 ≤ 𝑥) |
28 | | breq2 5057 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑎 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑎)) |
29 | 28 | rspccv 3534 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
𝐴 0 ≤ 𝑥 → (𝑎 ∈ 𝐴 → 0 ≤ 𝑎)) |
30 | 27, 29 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑎 ∈ 𝐴 → 0 ≤ 𝑎)) |
31 | 30 | imp 410 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 0 ≤ 𝑎) |
32 | 31 | adantrr 717 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 0 ≤ 𝑎) |
33 | | simp1r 1200 |
. . . . . . . . . . 11
⊢
(((∀𝑥 ∈
𝐴 0 ≤ 𝑥 ∧ ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) ∧ (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥)) → ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) |
34 | 11, 33 | sylbi 220 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 0 ≤ 𝑥) |
35 | | breq2 5057 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑏 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑏)) |
36 | 35 | rspccv 3534 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
𝐵 0 ≤ 𝑥 → (𝑏 ∈ 𝐵 → 0 ≤ 𝑏)) |
37 | 34, 36 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑏 ∈ 𝐵 → 0 ≤ 𝑏)) |
38 | 37 | imp 410 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 0 ≤ 𝑏) |
39 | 38 | adantrl 716 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 0 ≤ 𝑏) |
40 | | suprub 11793 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ sup(𝐴, ℝ, < )) |
41 | 12, 40 | sylan 583 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ sup(𝐴, ℝ, < )) |
42 | 41 | adantrr 717 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ≤ sup(𝐴, ℝ, < )) |
43 | | suprub 11793 |
. . . . . . . . 9
⊢ (((𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑦 ≤ 𝑥) ∧ 𝑏 ∈ 𝐵) → 𝑏 ≤ sup(𝐵, ℝ, < )) |
44 | 19, 43 | sylan 583 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ≤ sup(𝐵, ℝ, < )) |
45 | 44 | adantrl 716 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ≤ sup(𝐵, ℝ, < )) |
46 | 15, 18, 22, 25, 32, 39, 42, 45 | lemul12ad 11774 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
))) |
47 | 46 | ex 416 |
. . . . 5
⊢ (𝜑 → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎 · 𝑏) ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
)))) |
48 | | breq1 5056 |
. . . . . 6
⊢ (𝑤 = (𝑎 · 𝑏) → (𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, < )) ↔ (𝑎 · 𝑏) ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
)))) |
49 | 48 | biimprcd 253 |
. . . . 5
⊢ ((𝑎 · 𝑏) ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, < )) → (𝑤 = (𝑎 · 𝑏) → 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
)))) |
50 | 47, 49 | syl6 35 |
. . . 4
⊢ (𝜑 → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑤 = (𝑎 · 𝑏) → 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
))))) |
51 | 50 | rexlimdvv 3212 |
. . 3
⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑤 = (𝑎 · 𝑏) → 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
)))) |
52 | 10, 51 | syl5bi 245 |
. 2
⊢ (𝜑 → (𝑤 ∈ 𝐶 → 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
)))) |
53 | 52 | ralrimiv 3104 |
1
⊢ (𝜑 → ∀𝑤 ∈ 𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) · sup(𝐵, ℝ, <
))) |